In my opinion, it is not the towel, it is your early downswing. Even with the towel, you can still easily make out-in or in-out downswing through manipulation. This book martin recommended will tell you what the problem is:

This book was discussed in Golf Channel, School of Golf, chapter 28 recently. Martin Hall spoke highly of it. "fascinating stuff, very worth reading". You can search the title on youtube or golf channel.

Your left hand position on the top of the backswing has NOTHING to do with your swing plane. The club only moves on plane below your hips, on the top, it deviates from the plane, the difference between Woods and Kuchar (or Hogan) is how much the deviation is.
Your plane angle should be determined by the club lie angle, so with the same club you cannot change the steepness of your plane randomly. What you can change is how the club wraps around your body (higher left...

Very unique point of view on swing plane. If Jim Hardy has confused you with his swing theory, then you will probably find out why in this book. A "steep" two-plane swing is not necessarily steeper than a flat one-plane swing. To be honest, I have never been convinced by the swing plane theory from Jim Hardy.